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Register a new userSelf-attenuation of extreme events in Navier-Stokes turbulence
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Turbulent fluid flows are ubiquitous in nature and technology, and are mathematically described by the incompressible Navier-Stokes equations (INSE). A hallmark of turbulence is spontaneous generation of intense whirls, resulting from amplification of the fluid rotation-rate (vorticity) by its deformation-rate (strain). This interaction, encoded in the non-linearity of INSE, is non-local, i.e., depends on the entire state of the flow, constituting a serious hindrance in turbulence theory and in establishing regularity of INSE. Here, we unveil a novel aspect of this interaction, by separating strain into local and non-local contributions utilizing the Biot-Savart integral of vorticity in a sphere of radius R. Analyzing highly-resolved numerical turbulent solutions to INSE, we find that when vorticity becomes very large, the local strain over small R surprisingly counteracts further amplification. This uncovered self-attenuation mechanism is further shown to be connected to local Beltramization of the flow, and could provide a direction in establishing the regularity of INSE.
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We investigate the spatio-temporal structure of the most likely configurations realising extremely high vorticity or strain in the stochastically forced 3D incompressible Navier-Stokes equations. Most likely configurations are computed by numerically finding the highest probability velocity field realising an extreme constraint as solution of a large optimisation problem. High-vorticity configurations are identified as pinched vortex filaments with swirl, while high-strain configurations correspond to counter-rotating vortex rings. We additionally observe that the most likely configurations for vorticity and strain spontaneously break their rotational symmetry for extremely high observable values. Instanton calculus and large deviation theory allow us to show that these maximum likelihood realisations determine the tail probabilities of the observed quantities. In particular, we are able to demonstrate that artificially enforcing rotational symmetry for large strain configurations leads to a severe underestimate of their probability, as it is dominated in likelihood by an exponentially more likely symmetry broken vortex-sheet configuration.
We accomplish two major tasks. First, we show that the turbulent motion at large scales obeys Gaussian statistics in the interval 0 < Rlambda < 8.8, where Rlambda is the microscale Reynolds number, and that the Gaussian flow breaks down to yield place to anomalous scaling at the universal Reynolds number bounding the inequality above. In the inertial range of turbulence that emerges following the breakdown, the effective Reynolds number based on the turbulent viscosity, Rlambda* assumes this same constant value of about 9. This scenario works also for the emergence of turbulence from an initially non-turbulent state. Second, we derive expressions for the anomalous scaling exponents of structure functions and moments of spatial derivatives, by analyzing the Navier-Stokes equations in the form developed by Hopf. We present a novel procedure to close the Hopf equation, resulting in expressions for zetan in the entire range of allowable moment-order, n, and demonstrate that accounting for the temporal dynamics changes the scaling from normal to anomalous. For large n, the theory predicts the saturation of zetan with n, leading to two inferences: (a) the smallest length scale etan = LRe-1 << LRe-3/4, where Re is the large-scale Reynolds number, and (b) velocity excursions across even the smallest length scales can sometimes be as large as the large scale velocity itself. Theoretical predictions for each of these aspects are shown to be in quantitative agreement with available experimental and numerical data.
Whether the 3D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the 3D incompressible axisymmetric Navier-Stokes equations with smooth initial data of finite energy develop nearly singular solutions at the origin. This nearly singular behavior is induced by a potential finite time singularity of the 3D Euler equations that we reported in cite{Hou-euler-2021}. One important feature of the potential Euler singularity is that the solution develops nearly self-similar scaling properties that are compatible with those of the 3D Navier-Stokes equations. We will present numerical evidence that the 3D Navier-Stokes equations develop nearly singular scaling properties with maximum vorticity increased by a factor of $10^7$. Moreover, the nearly self-similar profiles seem to be very stable to the small perturbation of the initial data. However, the 3D Navier-Stokes equations with our initial data do not develop a finite time singularity due to the development of a mild two-scale structure in the late stage, which eventually leads to viscous dominance over vortex stretching. To maintain the balance between the vortex stretching term and the diffusion term, we solve the 3D Navier-Stokes equations with a time-dependent viscosity roughly of order $O(|log(T-t)|^{-3})$ in the late stage. We present strong numerical evidence that the 3D Navier-Stokes equations with such time-dependent viscosity develop a finite time singularity.
A conservative discretization of incompressible Navier-Stokes equations is developed based on discrete exterior calculus (DEC). A distinguishing feature of our method is the use of an algebraic discretization of the interior product operator and a combinatorial discretization of the wedge product. The governing equations are first rewritten using the exterior calculus notation, replacing vector calculus differential operators by the exterior derivative, Hodge star and wedge product operators. The discretization is then carried out by substituting with the corresponding discrete operators based on the DEC framework. Numerical experiments for flows over surfaces reveal a second order accuracy for the developed scheme when using structured-triangular meshes, and first order accuracy for otherwise unstructured meshes. By construction, the method is conservative in that both mass and vorticity are conserved up to machine precision. The relative error in kinetic energy for inviscid flow test cases converges in a second order fashion with both the mesh size and the time step.
We investigate the behaviour of a system where a single phase fluid domain is coupled to a biphasic poroelastic domain. The fluid domain consists of an incompressible Newtonian viscous fluid while the poroelastic domain consists of a linear elastic solid filled with the same viscous fluid. The properties of the poroelastic domain, i.e. permeability and elastic parameters, depend on the inhomogeneous initial porosity field. The theoretical framework highlights how the heterogeneous material properties enter the linearised governing equations for the poroelastic domain. To couple flows through this domain with a surrounding Stokes flow, we show case a numerical implementation based on a new mixed formulation where the equations in the poroelastic domain are rewritten in terms of three fields: displacement, fluid pressure and total pressure. Coupling single phase and multiphase flow problems are ubiquitous in many industrial and biological applications, and here we consider an example from in-vitro tissue engineering. We consider a perfusion system, where a flow is forced to pass from the single phase fluid to the biphasic poroelastic domain. We focus on a simplified two dimensional geometry with small aspect ratio, and perform an asymptotic analysis to derive analytical solutions for the displacement, the pressure and the velocity fields. Our analysis advances the quantitative understanding of the role of heterogeneous material properties of a poroelastic domain on its mechanics when coupled with a fluid domain. Specifically, (i) the analytical analysis gives closed form relations that can be directly used in the design of slender perfusion systems; (ii) the numerical method is validated by comparing its result against selected theoretical solutions, opening towards the possibility to investigate more complex geometrical configurations.
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